Finite generalized soluble groups

Let $\sigma =\{\sigma_{i} \mid i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. Suppose $\sigma (G)=\{\sigma_{i} \mid \sigma_{i}\cap \pi (G)\ne \varnothing\}$. A set $\mathcal{H}$ of subgroups of $G$ is called a complete Hall $\sigma $-set of $G$ if every nontrivial member of $\mathcal{H}$ is a $\sigma_{i}$-subgroup of $G$ for some $i\in I$ and $\mathcal{H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $i$ such that $\sigma_{i}\in \sigma (G)$. A group $G$ is $\sigma$-full if $G$ possesses a complete Hall $\sigma $-set. A complete Hall $\sigma $-set $\mathcal{H}$ of $G$ is called a $\sigma$-basis of $G$ if every two subgroups $A, B \in\mathcal{H}$ are permutable, i.e., $AB=BA$.
In this paper, we study properties of finite groups having a $\sigma$-basis. It is proved that if $G$ has a $\sigma$-basis, then $G$ is generalized $\sigma$-soluble, i.e, $|\sigma (H/K)|\leq 2$ for every chief factor $H/K$ of $G$. Moreover, every complete Hall $\sigma$-set of a $\sigma$-full group $G$ forms a $\sigma$-basis of $G$ iff $G$ is generalized $\sigma$-soluble, and for the automorphism group $G/C_{G}(H/K)$ induced by $G$ on any its chief factor $H/K$, we have $|\sigma (G/C_{G}(H/K))|\leq 2$ and also $\sigma(H/K)\subseteq \sigma (G/C_{G}(H/K))$ in the case $|\sigma (G/C_{G}(H/K))|= 2$.